Bennett Sorbo had an iPhone 6s with a dying battery. He ran his iPhone through some benchmarks, timing and filming the whole thing.
He then went to Apple, got a replacement battery, and ran the same tests again. The video below shows the results.
Bottom line, replacing the battery clearly speeds things up. If you jump to about 2:34 in, you’ll see that the tasks took 5:45 on the bad battery (a presumably throttled processor) and only 4:33 on the new battery.
That’s a savings of 1:12, or about 21%. Not necessarily accurate to say that your phone will be 21% faster with a new battery, but it certainly seems like a new battery would make your phone at least somewhat more nimble.